This paper is multifold in structure. At first, we further study closure continuity
of maps, a concept which was introduced in [2]. In specific, it is shown that restrictions
of such maps to subspaces maintain closure continuity, but not conversely. Then, strongly
open set topologies are defined. We found that, in general, this gives a weaker topology
but not in regular spaces where, the topologies turn out to be the same. It is also evident
that many traditional theorems hold for closure continuous maps acting on or into Uryson
Spaces. Just to mention, we prove that if f , g : X → Y are closure continuous functions
and Y is a Uryson Space, then:(i) the set {x∈ X : f (x) = g(x)} is strongly closed in X ,
(ii) The graph Gf = {(x, f (x)): x∈ X} of f is a strongly closed set in X ×Y . We also
prove that , When the space X is Uryson, and f : X →Y is closure continuous, then the
set E = {x∈ X : f (x) = x} of all fixed points under f is strongly closed in X . Finally,
we show that equivalent are: (i) X is a Uryson space (ii) Weak limits of filters in
X ,when exist are unique (iii) Weak limits of nets in X ,when exist are unique.(iν )The
diagonal ∆X = {(x, x): x∈ X}is a strongly closed set in the product space X × X . 

Title

Annals of Pure and Applied Mathematics

Publisher

House of Scientific Research

Publisher Country

India

Publication Type

Prtinted only

Volume

16

Year

2018

Pages

117-125